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First column in maze arrangement of natural numbers.
4

%I #14 Sep 08 2022 08:45:09

%S 1,2,3,12,13,30,31,56,57,90,91,132,133,182,183,240,241,306,307,380,

%T 381,462,463,552,553,650,651,756,757,870,871,992,993,1122,1123,1260,

%U 1261,1406,1407,1560,1561,1722,1723,1892,1893,2070,2071,2256,2257,2450,2451

%N First column in maze arrangement of natural numbers.

%C Interleaves two times the hexagonal numbers A000384 with A054554.

%H Vincenzo Librandi, <a href="/A081347/b081347.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(n) = ((1+2*n^2)+(1-2*n)*(-1)^n)/2.

%F a(2n) = A054554(n).

%F a(2n+1) = 2*A000384(n).

%F G.f.: (1+x-x^2+7*x^3)/((1-x)^3*(1+x)^2). [_Colin Barker_, Apr 17 2012]

%e Starting with 1,2,3, turn (LL) and then repeat (RRR)(LLL) to get

%e 1 6 7 20

%e 2 5 8 19

%e 3 4 9 18

%e 12 11 10 17

%t CoefficientList[Series[(1 + x - x^2 + 7 x^3) / ((1 - x)^3 (1 + x)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 08 2013 *)

%o (Magma) [((1+2*n^2)+(1-2*n)*(-1)^n)/2: n in [0..50]]; // _Vincenzo Librandi_, Aug 08 2013

%Y Cf. A081348, A080335.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 19 2003